A000113 Number of transformation groups of order n.

1, 3, 4, 3, 6, 12, 8, 6, 4, 18, 12, 12, 14, 24, 24, 6, 18, 12, 20, 18, 32, 36, 24, 24, 30, 42, 12, 24, 30, 72, 32, 12, 48, 54, 48, 12, 38, 60, 56, 36, 42, 96, 44, 36, 24, 72, 48, 24, 56, 90, 72, 42, 54, 36, 72, 48, 80, 90, 60, 72, 62, 96, 32, 12, 84, 144, 68

Offset

1,2

Comments

A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Jan 03 2017

References

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 139.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Index to divisibility sequences

Formula

Let psi(m) = A001615(m) (Dedekind's psi function). Write n = 2^i*3^j*k, where (6,k) = 1 and let i' = floor(i/2) for i < 6, i' = 3 for i >= 6; let j' = 0 for j = 0 or 1, j' = 1 for j >= 2. Then a(n) = psi(n/(2^i'*3^j')) = psi(n)/(2^i'*3^j').

Multiplicative with a(2^e)=3*2^Floor[(e-1)/2] for 0<e<7, a(2^e)=3*2^(e-4) for e>=7, a(3^e)=4 for 0<e<3, a(3^e)=4*3^(e-2) for e>=3 and a(p^e)=(p+1)*p^(e-1) for p>3. - T. D. Noe, Nov 14 2006

Sum_{k=1..n} a(k) ~ c * n^2, where c = 43501/(7680*Pi^2) = 0.573902... . - Amiram Eldar, Oct 23 2022

Example

G.f. = x + 3*x^2 + 4*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 6*x^8 + 4*x^9 + ...

Mathematica

psi[n_] := n*DivisorSum[n, MoebiusMu[#]^2/#&]; a[n_] := (i=IntegerExponent[ n, 2]; j=IntegerExponent[n, 3]; ip = If[i<6, Floor[i/2], 3]; jp = If[j<2, 0, 1]; psi[n]/(2^ip*3^jp)); Array[a, 60] (* Jean-François Alcover, Feb 04 2016 *)
a[ n_] := If[ n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors @ n}] / (2^Min[3, Quotient[IntegerExponent[n, 2], 2]] 3^Boole[1 < IntegerExponent[n, 3]]) ]; (* Michael Somos, Jan 03 2017 *)

Prog

(PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, moebius(d)^2 / d) / (2^min(3, valuation(n, 2)\2) * 3^(1 < valuation(n, 3))))}; /* Michael Somos, Jan 03 2017 */

Crossrefs

Cf. A001615.

Sequence in context: A109506 A369889 A365481 * A365347 A069915 A033634

Adjacent sequences: A000110 A000111 A000112 * A000114 A000115 A000116

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved